Based off the video, you’ve got some technique tweaks we can apply. First- you’ve got the bar set a bit too high. So when you take it out of the rack you’re lifting with your delts by pressing up. What would be best is if you can tricep extension out of the rack and pull it over with your lats.
Lones has a solid video on this:
Then when you’re holding the bar before the press, you’ve got the bar a bit closer to your face almost at a slight backwards incline. Move the bar just a bit further out over your chest so the barbell sits in line with your shoulder. ( if your shoulders are retracted back this will allow all the weight to sit in your back and not in your shoulders/chest )
When you touch the bar to your chest, your elbows end up behind the barbell, this means that when you go to press, you have to tuck the elbows hard to get them back under the barbell to create upward momentum. The solution to this is to tuck harder or touch slightly higher so the elbows stay in line with the barbell.
Your decent looks solid as far as speed is concerned, but when you touch your chest you are relaxing, stay tight and push yourself away from that bar as hard as you can.
Supplemental some 5s negatives to teach you maintaining tightness or some spoto presses could do wonders.
For accessory focus I’d say work on your lats, rear delts, rhomboids and some higher rep DB work ( naturally keep in your triceps, lateral raises, etc, but focus on the above )
With some small tweaks I think you’ll notice your bench going up.